On Borel almost disjoint families

Abstract

There is a close correspondence between uncountable almost disjoint families of subsets of $$\omega $$ and Aleksandrov–Urysohn compacta (in short, AU-compacta)—separable, uncountable compact spaces whose second derived set is a singleton. We shall show in particular, that AU-compacta embeddable in the space of first Baire class functions on the Cantor set $$2^\omega $$ , with the pointwise topology, are exactly the ones determined by almost disjoint families that are Borel sets in $$2^\omega $$ , and they are also distinguished among AU-compacta by the property that the cylindrical $$\sigma $$ -algebras of their function spaces are standard measurable spaces. Although the first condition implies the third one for arbitrary separable compact space, it is an open problem, whether the reverse implication is always true.